Title: Superconvergence of Discontinuous Galerkin methods for linear hyperbolic equations

Speaker: 曹外香,北京师范大学

Time: Nov. 28

Abstract: In this talk, we will introduce superconvergence properties of discontinuous Galerkin (DG) methods linear hyperbolic conservation laws when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the ($2k+1$)-th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of ($k+1$)-th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of ($k+2$)-th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.

Title: Well-posedness of the initial value problem for a class of time-dependent partial integro-differential equations

Speaker: 卢朓, 北京大学

Time: Nov. 21

Abstract: The abstract initial value problem for the system of evolution equations for a real-valued function and a function-valued function was considered. The existence and uniqueness of classical solution to the evolution system were proved in a Banach space under assumptions on the boundedness and smoothness of data. An isomorphism between the solution space of the evolution system and a special subspace of the Schwartz space is established. It is verified that the solution of the evolution system can be mapped to a function which is the solution of the initial value problem of an integral-differential equation. The theoretical finding has a potential application in studying the well-posedness of the stationary Wigner equation with inflow boundary conditions.

Title: Energy stable local discontinuous Galerkin methods

Speaker: Li Guo, 中山大学

Time: Nov. 14

Abstract: Energy stable local discontinuous Galerkin (LDG) method and its two applications have been investigated in this talk, including the Keller-Segel (KS) chemotaxis model and the nonlinear Schrodinger equation with wave operator (NLSW). The KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. A free energy which is decreasing during time evolution has been constructed. Even though the energy can be negative, it is always positive if the blow-up will occur. Meanwhile, the NLSW problem also has a conservative energy which is important to simulate long time behavior and eliminate oscillations. In this talk, we will construct a special energy stable LDG method to approximate the KS chemotaxis model with blow-up solution and numerically preserve the energy as well as a fully discrete energy conserving scheme utilizing the LDG method in space and the Crank-Nicholson algorithm in time to simulate the NLSW problem. Some numerical experiments for these two problems will be given to demonstrate the validity and performance of the energy stable LDG method.

Title: Convergence of discrete Aubry-Mather model in the continuous limit and related topics

Speaker: Xifeng Su, 北京师范大学

Time: Nov. 2

Abstract: We will consider the Frenkel-Kontorova models and their higher dimensional generalizations and talk about the corresponding discrete weak KAM theory. The existence of the discrete weak KAM solutions is related to the additive eigenvalue problem in ergodic optimization. In particular, I will show that the discrete weak KAM solutions converge to the weak KAM solutions of the autonomous Tonelli Hamilton-Jacobi equations as the time step goes to zero. This is a joint work with Prof. Philippe Thieullen.

Title: CENTRAL-UPWIND SCHEMES FOR SHALLOW WATER MODELS

Speaker: Alexander Kurganov, 南方科技大学

Time: Oct. 24

Abstract: In the first part of the talk, I will describe a general framework for designing finite-volume methods (both upwind and central) for hyperbolic systems of conservation laws. I will focus on Riemann-problem-solver-free non-oscillatory central schemes and, in particular, on central-upwind schemes that belong to the class of central schemes, but has some upwind features that help to reduce the amount of numerical diffusion typically present in staggered central schemes such as, for example, the first-order Lax-Friedrichs and second-order Nessyahu-Tadmor scheme. In the second part of the talk, I will discuss how central-upwind schemes can be extended to hyperbolic systems of balance laws, such as the Saint-Venant system and related shallow water models. The main difficulty in this extension is preserving a delicate balance between the flux and source terms. This is especially important in many practical situations, in which the solutions to be captured are (relatively) small perturbations of steady-state solutions. The other crucial point is preserving positivity of the computed water depth (and/or other quantities, which are supposed to remain nonnegative). I will present a general approach of designing well-balanced positivity preserving central-upwind schemes and illustrate their performance on a number of shallow water models.

Title: Parabolic Transmission Problems on Smooth and Polygonal Domains and Application to Finite Element Method with Graded Mesh

Speaker: Yajie Zhang, YMSC

Time: Oct. 17

Abstract: We study theoretical and practical issues of the second-order parabolic equation u_t +Lu = f, where L = −div(A∇) is a second-order operator with piecewise smooth coefficient matrix A, with possibly jump discontinuities across a finite number of curves, called the interface. First we concentrate on the problems with certain homogeneous or non-homogeneous boundary and interface conditions on smooth domain Ω with smooth interface Γ . Afterwards we analyze the problem on polygonal domains. Under some additional conditions we establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, or when the interfaces are not smooth, we fail to acquire well-posedness on weighted Sobolev space but we are able to obtain the decomposition u = ureg + ws, into a function ureg with better decay at the vertices and a function ws that is locally constant near the vertices, thus proving well-posedness in an augmented space. Based on the theoretical analysis we are able to implement a certain Finite Element scheme with improved graded meshes, which can recover the rate of convergence for piecewise polynomials of degree m ≥ 1. Three numerical tests are included in the last.

Title: Staggered Discontinuous Galerkin Methods for Stokes problem and elastodynamics

Speaker: Jie Du, YMSC

Time: Oct. 10

Abstract: Staggered discontinuous Galerkin (SDG) method is a new class of discontinuous Galerkin (DG) methods, which uses staggered mesh. It combines some good properties of finite element methods and standard DG methods through the use of staggered grid. In this talk, we develop an adaptive SDG method for the Stokes system. A computable error indicator is constructed and its reliability and efficiency are proved. Moreover, by combing the features of SDG method and traditional hybridization method, we present a staggered hybridization technique for DG methods to discretize linear elastodynamic equations. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error.

Title: Self-Organized Hydrodynamic models for nematic alignment and the application to myxobacteria

Speaker: Hui Yu, YMSC

Time: Sep. 26

Abstract: A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or antiparallel to this mean direction. An application to myxobacteria is presented.

Title: Coordinate Descent Methods

Speaker: Wotao Yin, UCLA

Time: Sep. 18 (周一下午3:15), Lecture Room 1, Floor 1, Jin Chun Yuan West Building

Abstract: This talk overviews a class of algorithms called coordinate descent algorithms and also discusses its recent progress. This class of algorithms has recently gained popularity due to their effectiveness in solving large-scale optimization problems in machine learning, compressed sensing, and image processing. Coordinate descent algorithms solve optimization problems by successively minimizing along each coordinate, or block of coordinates, which is ideal for parallelized and distributed computing. This talk gives relevant theory and examples about how to effectively apply coordinate descent to modern problems in data science and engineering, how to linearly speed up the algorithm by asynchronous parallel computing, and how to obtain global optimality guarantees from those on each coordinate.